Lebesgue Integral converging and integrable sequences

Question

Have a problem with second part of the question

Assume $f_n,g_n \in L^1(\mathbb{R},\mu)$, $f_n \rightarrow 0$ and $g_n \rightarrow 0$ a.e., as $n \rightarrow \infty$. Prove

$$ \lim_{n\rightarrow\infty} \int_A \frac{2f_n(x)g_n(x)}{1+g_n^2(x)+f_n^2(x)}d\mu(x) = 0 $$

for any set $A \subset \mathbb{R}$ of finite measure. Show by example that this does not extend to the whole $\mathbb{R}$.

Solution Let $h_n = \frac{2f_ng_n}{1+f_n^2+g_n^2}$ we have that $0 \leq h_n \leq g_n$. Further inspection even shows that $0 \leq h_n \leq 1$ and $h_n \rightarrow 0$ so on a set of finite measure the integral goes to 0. How to show this isn't the case if the set of measure isn't finite?

Answer 1

If $A=\mathbb R$ and $f_n=g_n=\mathbf 1_{(n,2n)}$, then $$ \int_A \frac{2f_n(x)g_n(x)}{1+g_n^2(x)+f_n^2(x)}d\mu(x) =\frac 23\mu(n,2n)=2n/3. $$

Answer 2

One possibility is $f_n=g_n=\mathbb{1}_{(n,n^2]}$ in which case \begin{aligned} \frac{2f_ng_n}{1+f^2_n+g^2_n}\geq \frac{1}{2}f_n \end{aligned} With respect Lebesgue measure \begin{aligned} \int f_n =n^2-n\rightarrow\infty \end{aligned}